In this paper, we extensively investigate the way in which $κ$-$μ$ fading channels can be impacted by shadowing. A family of shadowed $κ$-$μ$ fading models are introduced and classified according to whether the underlying $κ$-$μ$ fading undergoes single or double shadowing. We discuss three types of single shadowed $κ$-$μ$ model (denoted Type I to Type III) and three types of double shadowed $κ$-$μ$ model (denoted Type I to Type III). The taxonomy of the single shadowed Type I - III models is dependent upon whether the fading model assumes that the dominant component, the scattered waves, or both experience shadowing. The categorization of the double shadowed Type I - III models is dependent upon whether a) the envelope experiences shadowing of the dominant component, which is preceded (or succeeded) by a secondary round of shadowing (multiplicative), or b) the dominant and scattered contributions are fluctuated by two independent shadowing processes, or c) the scattered waves of the envelope are subject to shadowing, which is also preceded (or succeeded) by a secondary round of multiplicative shadowing. Although the physical definition of the examined models make no predetermination of the statistics of the shadowing process, for illustrative purposes, two example cases are provided for each type of single and double shadowed model by assuming that the shadowing is shaped by a Nakagami-$m$ random variable (RV), an inverse Nakagami-$m$ RV or their mixture. The double shadowed $κ$-$μ$ models offer remarkable flexibility as they include the $κ$-$μ$, $η$-$μ$, and the various types of single shadowed $κ$-$μ$ distribution as special cases. Moreover, we demonstrate a practical application of the double shadowed $κ$-$μ$ Type I model by applying it to channel measurements obtained for body area networks operating at 2.45 GHz.